spectral scaling analysis of rr lyrae stars ... - vivek kohar · 2 vivek kohar, john f. lindner,...

Spectral scaling analysis of RR Lyrae stars inOGLE-IV Galactic bulge fields

Vivek Kohar, John F. Lindner, Behnam Kia and William L. Ditto

Abstract Recent studies of variable stars have uncovered characteristic nonlinearfeatures in flux of these stars and indicate the presence of quasiperiodicity. A com-mon technique to study quasiperiodic systems is spectral scaling analysis whichrelies on the fact that different dynamical behaviors can be identified on the basis ofdistribution of peaks in the periodogram. Here we apply the spectral scaling tech-nique to the OGLE-IV photometry of the RR Lyrae stars in the Galactic bulge. Wefind that spectra of the fundamental mode (RRab) and first overtone RR Lyrae stars(RRc) scales differently and thus the spectral scaling can be used to distinguish be-tween different RR Lyrae sub classes. Furthermore, goodness of fit for RRc starswith multiple modes is better than other stars. The scaling exponent for stars ob-served in high cadence is close to the values reported using Kepler photometry. Thisanalysis can help us to reclassify the stars based on their dynamical characteristics.

1 Introduction

Recent high resolution photometry of RR Lyrae variable stars has shown that theirdynamics is much more complex than previously thought and the Blazhko effect isnot the only unsolved puzzle about their complex behavior [1, 2, 3, 4, 5, 6, 7]. Spec-tral analysis of the Kepler and Optical Gravitational Lensing Experiment (OGLE)photometry has shown that their are many additional excited modes. Blazhko effectand period doubling was observed in many RR Lyrae stars in Kepler field of view[8]. Molnar et. al [9] report similar findings in variable stars observed with K2 mis-

Vivek Kohar, Behnam Kia and William L. DittoDepartment of Physics, North Carolina State University, Raleigh, North Carolina 27695-8202 USAe-mail: [email protected]

John F. LindnerPhysics Department, The College of Wooster, Wooster, Ohio 44691 USA

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2 Vivek Kohar, John F. Lindner, Behnam Kia and William L. Ditto

sion [9, 10]. The precise origin and nature of these additional modulations and theirpossible connection to Blazhko effect is yet to be fully understood.

Switching of dominant pulsation mode has been reported in some RR Lyrae stars[11]. Bryant [12] has proposed that Blazhko effect is due to interaction of near res-onant modes [12, 13]. Lindner et. al [14] proposed that the dynamics of RRc starsin Kepler field of view is strange nonchaotic implying that their light intensity has afractal attractor. All these studies highlight that nonlinearity plays a significant rolein the dynamics of these stars and is being incorporated in hydrodynamic models ofstellar pulsation [15].

The classification of RR Lyrae stars in sub classes plays an important role inasterioseismology and numerous methods have been proposed for the same [16,17, 18, 19, 20]. Our method of spectral scaling complements these approaches andcan be used to distiguish other subtle characteristics which may be same or differentin these subclasses.

2 Data and methods

We used the publicly available data from OGLE−IV project 1 [21]. Soszynski etal. [22] reported more than 38257 RR Lyrae stars in the Galactic bulge as observedduring OGLE−IV 2. These include about 27258 RRab, 10825 RRc stars and 174RRd stars. Few of these stars have been reclassified. Most of the observations inOGLE IV were in I-band and the number of collected epochs varies from 80 tomore than 8000 [22]. Hence we used the I-band data for our analysis.

2.1 Methodology

Accurate periodogram computation for astronomical time series is difficult as thedata points are noisy and unevenly sampled. For more details, refer to [23] andreferences therein. Many phenomenon in nature exhibit scaling indicating an under-lying relationship between the variables [e.g., see [24, 25, 14, 26, 27]]. Lindner et.al [26] reported that the spectra of RRc stars in Kepler field of view scales with anexponent close to −1.5 and we perform similar analysis for this database.

We calculated the Lomb-Scargle periodogram using FNPEAKS 3 for frequenciesbetween 10−4 to 24 with a resolution of 10−4. Then we identified the peaks in theperiodogram and sorted them in descending order of amplitude (SNR) of the peaks.We selected a fixed number of the peaks starting from the highest peak and plottedthe peak height vs peak number. Then we fitted a linear polynomial on log−log scale

1 http://ogle.astrouw.edu.pl2 ftp://ftp.astrouw.edu.pl/ogle/ogle43 http://helas.astro.uni.wroc.pl

Spectral scaling analysis of RR Lyrae stars in OGLE-IV Galactic bulge fields 3

and called it a ”scaling exponent”. We also calculated the 95 percent confidenceintervals and adjusted r−square indicating the goodness of fit. We performed thisanalysis for all the 38000 stars for different number of peaks and SNR values. Agood scaling does not necessarily imply a power law [28] and our primary goal inthis manuscript is to classify the RR Lyrae variables based on scaling exponent.The scaling exponent and shape of folded light curves are closely related. As theshape of the curve distorts from sinusoidal, more and more frequency componentsare needed to generate the curve.

2.2 Generality

To evaluate the generality of our results we tested the results of a small sample ofstars by calculating the periodogram with PERIOD04 and obtained similar results.We also verified the results by taking a small interval of frequencies from 10−4 to10 days.

3 Results

Large number of data points is vital to the study of variable stars and extraction ofnonlinear features. So we first identified stars with more than 8000 data points inthe I −band. First we present the analysis of the stars for which high cadence datais available followed by stars with lesser number of data points. The number of datapoints affects the scaling exponent. The typical data for a star is shown in Fig 1.We observe that there are large gaps in the time series which correspond to activeand inactive observation periods. This leads to strong aliases at one year period. Forstars with few data points, these aliases dominate the periodogram.

The corresponding LS periodogram for the time series shown in Fig. 1 is shownin Fig. 2. We observe that periodogram is quite rough and there are many peakscorresponding to higher harmonics as well as other frequency combinations. Pres-ence of multiple harmonics suggests non sinusoidal light curves. Other frequenciesindicate aperiodic behavior. The lower panel pots the scaling of the periodogram.We observe that it scales with an exponent close to −1.5 which is similar to whatwas observed in Kepler data for RRc stars.

Next we plot the periodogram of a RRab star in Fig. 3. Similar to RRc stars,there is significant power in other harmonics and daily aliases. The slope of linefitted on log−log scale shown in the bottom panel of Fig. 3 indicates a spectralscaling exponent close to −2. This indicates that power at other frequencies fall offrapidly in RRab stars as compared to RRc stars or the periodogram of RRc stars ismore rough compared to RRab stars.

We repeated the same analysis for all the RR Lyrae stars and the scaling exponentfor RRab and RRc stars with more than 8000 data points is plotted in Fig. 4. We


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Fig. 1 The time series of RRc star OGLE-BLG-RRLYR-05301. Note the seasonality in the avail-ability of data points.

observe that RRab and RRc stars are clearly separated in terms of scaling exponent.Thus the phenomenon of roughness of periodogram of RRc stars is valid in general.Secondly the stars form a very prominent cluster which suggests that this type ofscaling is a distinguishing feature of these stars. We observed similar scaling lawfor RRc stars using Kepler data. Both long cadence and short cadence data fromKepler gave an exponent of −1.5. As the number of data points in Kepler data wasmuch higher, it indicates that for sufficiently large number of data points we can geta good estimate of the scaling exponent which saturates near −1.5.

Netzel et. al [29] reported that about 27 percent of high cadence stars in OGLEIV data of galactic bulge show a mode near period ratio of (0.60− 0.64). We thenselected this particular sample of stars and in Fig. 4 we plot the scaling exponentfor this subset of stars. We observe that the stars with additional modes form avery close cluster with a scaling exponent of −1.5 consolidating our hypothesis thatadditional modes are reflective of singular spectra.

Next we studied the impact on scaling exponent when the number of data pointsis low. When the number of data points decreases, the SNR of peaks decreases andwe get very few peaks which are above our SNR ≥ 10 threshold. For some reason-able scaling we need the number of peaks to be more than 10 and so we loweredour threshold to SNR ≥ 4 and the corresponding scaling exponents are shown inFig. 5. The periodogram of OGLE data has strong 1 day aliases and these corruptour analysis for lower SNR values. If we decrease the SNR threshold, there is higherlikelihood that peaks due to noise and aliases will also be counted in our analysis.We observe that when we lower the SNR threshold then the noise and aliasing termsbecome dominant and both RRab and RRc have similar scaling exponent though


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Fig. 2 The LS periodogram of OGLE-BLG-RRLYR-05301. Peaks with SNR ≥ 10 have beenmarked. Day and night cycle introduce additional aliases. The scaling of the number of peaksabove a certain threshold versus the threshold is shown on the figure below. A line of slope −1.5is shown as a guide to eye.


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Fig. 3 LS periodogram and scaling exponent of a RRab star OGLE-BLG-RRLYR-03599. Peakswith SNR ≥ 10 have been marked. A line of slope −2 is fitted to the data.

the goodness of fit is better for RRab stars and the two clusters of stars are stilldistinguishable. Note that the Kepler database had more than 60000 data points inlong cadence and even more for short cadence and the scaling exponents were stillsimilar for peaks with SNR ≥ 10. This indicates that the scaling exponents saturatefor sufficiently large number of data points so that peaks due to noise and aliases donot contribute in our analysis.


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Fig. 4 The scaling exponent of RRab and RRc stars plotted against goodness of fit for peakswith SNR ≥ 10 (left). The scaling exponent of RRc golden stars identified in [29] for peaks withSNR ≥ 10 (right).

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Fig. 5 The scaling exponent of RRab and RRc stars plotted against goodness of fit for peaks withSNR ≥ 4.

Next we studied the variation of scaling exponent with number of data pointsfor a fixed SNR. In Fig. 7 we plot the scaling exponent for all the RR Lyrae starsin OGLE−IV Galactic bulge database. We find that low SNR successfully distin-guishes RRab and RRc stars for less number of data points but as the number of datapoints increases, the aliasing peaks become dominant and we need to select higherSNR values to obtain the scaling relation. The number of RRd stars is still smallin the database and it is difficult to arrive at concrete generalization but the limiteddata suggest that they are distributed between the RRab and RRc stars.


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Fig. 6 The scaling exponent and goodness of fit of RRab(blue), RRc (gold) and RRd (green) starswith number of data points for peaks with SNR ≥ 4.

We also studied relation of scaling exponent with dominant period as shown inFig. 7. We can infer that the periodogram of RRab stars becomes rougher as periodincreases. The scaling exponent for RRc stars doesn’t exhibit any consistent pattern.

We further studied if the Fourier parameters are related with scaling exponent.The results are shown in Fig. 8


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Fig. 7 The scaling exponent plotted against the dominant period for RRab and RRc stars withmore than 8000 data points and peaks with SNR ≥ 10

4 Conclusion

We have studied the scaling of periodogram go RR Lyrae stars in OGLE−IV Galac-tic bulge database. We found that RRc and RRab stars scale differently indicatingthat the periodogram of RRc stars is rougher compared to RRab stars. This analy-sis helps us to reclassify the stars in various sub classes based on their dynamicalcharacteristics. As evident from the analysis presented here, the dynamical charac-


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Fig. 8 The Fourier parameters (y axis) plotted against the scaling exponent (x axis) for RRab (blue)and RRc (gold) stars with more than 8000 data points and peaks with SNR ≥ 10

teristics closely overlap with the existing classification which is primarily based onthe shape of folded light curves. The spectral scaling does depend on factors like thecharacteristic non sinusoidal oscillations of RRab and RRc, number of data pointsand the specific SNR threshold elected for cut off peaks but for sufficiently largenumber of data points the results are robust and similar to results obtained fromKepler database.


Acknowledgements We gratefully acknowledge support from the Office of Naval Research underGrant No. N000141-21-0026 and STTR grant No. N00014-14-C-0033.

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spectral scaling analysis of rr lyrae stars ... - vivek kohar · 2 vivek kohar, john f. lindner,...

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